MATH 133 – Calculus with Fundamentals 1 Discussion Day on Lines and Linear Functions September 8, 2015

Background

Every line in the plane is described by an equation of the Ax + By + C = 0 for some constants A, B, C. If the line is not vertical (B ≠ 0), then it is the graph of a function. Recall that we say a function f is linear if f(x) = mx + b for some constants m, b. The number m is called the slope of the line and the constant b is called the y-intercept of the line.

Questions

1) Consider lines with equations of the form 2x + cy − 3 = 0.

(a) For which value of c does the line contain the point (1, 2)?

(b) For which value of c does the line have slope −5?

(c) Is there any value of c such that the line is horizontal? Why or why not?

(d) For which value of c is the line perpendicular to the line given by 5x−3y +1 = 0? (Hint: What is true about slopes of perpendicular lines?)

2) Many materials, including metal rods, expand when they are heated. Consider a steel rod that has length L_{0} (centimeters) when the temperature is T_{0} degrees Celsius. If the temperature is increased by ΔT degrees Celsius, the length of the rod will change by (approximately)

ΔL = αL_{0}ΔT,  (1)

where α = 1.24 × 10^{−5} (and ΔT is assumed to be not too large or too small). The constant α in (1) is called the thermal expansion coefficient of steel (the units of α are \frac{1}{degrees C}).

(a) A steel rod has length L_{0} = 40 cm at T_{0} = 40 degrees C. What will the length be at 90 degrees C?

(b) Find the length of the rod at T = 50 degrees C if the temperature at T_{0} = 100 degrees C is 65 cm.

(c) Express the length L as a function of T if L_{0} = 65 cm at T_{0} = 100 degrees C.

(d) Explain why (1) expresses L as a linear function of T for any given L_{0}, T_{0}.

3) The volume V (in liters) of sample of 3 grams of carbon dioxide at 27 degrees Celsius was measured as a function of the pressure p (in atmospheres) with the results in the following table:

\begin{matrix}p & 0.25 & 1.00 & 2.50 & 4.00 & 6.00 \\ V & 6.72 & 1.68 & 0.67 & 0.42 & 0.27 \end{matrix}

Is V (approximately) a linear function of p? Why or why not? If so, find an approximate formula V = mp + b. If not, can you see a equation of a different form for V as a function of p?